Frito-Lay - Logistics Model

Vol. 39, No. 5, September–October 2009, pp. 460–475issn 0092-2102 �eissn 1526-551X �09 �3905 �0460

informs ®

doi 10.1287/inte.1090.0450©2009 INFORMS

An Integrated Outbound Logistics Model forFrito-Lay: Coordinating Aggregate-LevelProduction and Distribution Decisions

Sıla Çetinkaya, Halit ÜsterDepartment of Industrial and Systems Engineering, Texas A&M University, College Station, Texas 77843

{[email protected], [email protected]}

Gopalakrishnan EaswaranDepartment of Engineering, School of Science, Engineering and Technology, St. Mary’s University,

San Antonio, Texas 78228, [email protected]

Burcu Baris KeskinDepartment of Information Systems, Statistics, and Management Science, University of Alabama,

Tuscaloosa, Alabama 35487, [email protected]

In this paper, we describe research to improve Frito-Lay’s outbound supply chain activities by simultaneouslyoptimizing its inventory and transportation decisions. Motivated by Frito-Lay’s practice, we first develop amixed-integer programming formulation from which we develop a large-scale, integrated multiproduct inven-tory lot-sizing and vehicle-routing model with explicit (1) inventory holding costs, truck loading and dispatchcosts, and mileage costs; (2) production, storage, and truck capacity limitations; and (3) direct (plant-to-store)and interplant (plant-to-plant) delivery considerations. Second, we present an iterative solution approach inwhich we decompose the problem into inventory and routing components. The results demonstrate the impactof direct deliveries on distribution costs and show that direct deliveries and efficient inventory and routingdecisions can provide significant savings opportunities over two benchmark models, one of which representsthe existing Frito-Lay system. We implemented our models using an application that allows strategy evaluation,analysis of output files, and technology transfer. This application was particularly useful in evaluating potentialdirect-delivery locations and inventory reductions throughout the supply chain.

Key words : coordinated logistics; integrated inventory; transportation decisions; vendor-managed inventoryand delivery; inventory lot sizing; vehicle routing.

History : Published online in Articles in Advance June 10, 2009.

Frito-Lay North America (FLNA) operates a largeand complex supply chain. The company, a divi-

sion of PepsiCo, employs more than 40,000 peopleand produces a wide variety of snack foods. It ownsa private fleet, which consists of more than 1,400over-the-road (OTR) trucks and 20,000 route-delivery(RD) trucks of various capacities, for the outboundtransport of these products to many delivery loca-tions; these include distribution centers (DCs), bins,and customers (stores). Its overall distribution oper-ation also includes interplant (plant-to-plant) ship-ments because each plant does not produce eachproduct group. FLNA’s offerings comprise 16 differ-ent groups of produced and nonproduced items, withthousands of stock-keeping units (SKUs). It operates

33 production facilities, 192 DCs, and 1,535 bins in theUnited States as well as additional plants, DCs, andbins in Canada. Consequently, FLNA faces complexoutbound logistics planning issues.FLNA participates in a vendor-managed inven-

tory and delivery (VMI/D) program. Under a typicalVMI/D program, a vendor is empowered to con-trol its resupply timing and quantity at downstreamlocations (Çetinkaya and Lee 2000). Because effec-tively utilizing transportation resources is imperative,a vendor is more likely to dispatch full-vehicle out-bound loads to achieve economies of scale. Usingthis program also makes coordinating inventory andoutbound transportation decisions easier. We devel-oped a model to allow simultaneous optimization of

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Çetinkaya et al.: An Integrated Outbound Logistics Model for Frito-LayInterfaces 39(5), pp. 460–475, © 2009 INFORMS 461

these decisions to improve outbound supply chainefficiency and, thus, enhance the success of FLNA’sVMI/D program.We focused on the outbound supply chain of a

major FLNA plant in Irving, Texas, and developedan integrated cost-minimization model that links pro-duction output and shipment quantities destined todelivery locations; thus, it allows demand to be sat-isfied in a cost-effective and timely manner. Themodel is a mixed-integer programming formulationthat seeks to achieve the following:(1) Inventory optimization and reduction at the

Irving plant’s warehouse (called the factory ware-house henceforth);(2) Load optimization for transportation from the

factory warehouse to the delivery locations (i.e., DCs,bins, plants, and direct-delivery (DD) customers),while allowing direct (plant-to-store) and interplantshipments; and(3) Flow optimization throughout the entire out-

bound supply chain.Our model’s uniqueness is its ability to coordinate

inventory and shipment quantities with truck load-ing and routing decisions. The model also considersthe option that DD customers can receive suppliesdirectly from the factory warehouse, DCs, and bins.Hence, we also seek to identify the DD customers ina customer set and measure the value of direct deliv-eries to them.The problem we address is a large-scale, integrated

multiproduct inventory lot-sizing and vehicle-routingproblem with explicit inventory holding costs, truckloading and dispatch costs, and mileage costs; pro-duction, storage, and truck capacity limitations; anddirect and interplant delivery considerations. It isa challenging generalization of the vehicle-routingproblem, which is NP-hard. Therefore, an exact opti-mal solution is impractical. We also cannot obtain asolution using existing software and FLNA’s real data.We overcome these difficulties by developing a newheuristic that decomposes the overall problem intoinventory and routing subproblems.In the inventory subproblem, we determine the

final set of DD customers and the weekly replenish-ment and shipment quantities to the delivery loca-tions while minimizing relevant costs. In the routingsubproblem, we find the subsets of delivery locations

that trucks dispatched from the factory warehousemust visit. In this subproblem, we utilize the weeklyreplenishment and shipment quantities, which weobtain from solving the inventory subproblem, as dis-tribution requirements at the delivery locations. Theobjective of the routing subproblem is to minimizeloading and routing costs and maximize truck uti-lization. Clearly, convergence might occur to a localoptimum, and it is impossible to benchmark solutionquality relative to the global optimum. To demon-strate the effectiveness of our optimization model andproposed solution approach, we compare our resultsto two practical benchmarks, the chase policy and thecube-out policy (see descriptions in Benchmark Modelsand Benefits Realized). The former is of interest to thecompany for policy comparison purposes; the latterrepresents its usual practice. The solution approachfor the benchmark models relies on the solution thatwe developed for our model. Using numbers, weshow that run times in our solution approach arereasonable, and our model provides significant costsavings.We organize the remainder of the paper as follows.

We present a brief review of the relevant literaturein Related Literature. In Problem Setting and Formula-tion, we develop a mathematical formulation of theproblem. In Solution Methodology, we discuss prob-lem properties to show the difficulties of developingsolutions for our problem; we also present our pro-posed approach. We describe the software applicationthat combines the database and the solution approachin Application Modules. In Benchmark Models and Bene-fits Realized, we discuss our proposed model’s perfor-mance and demonstrate that our approach produceshigh-quality solutions when compared to the bench-mark models. Finally, we summarize our findings inConclusions.

Related LiteratureOur problem deals with integrated logistical planningwith a specific emphasis on coordinating inventoryand transportation decisions. A large body of quan-titative research exists to address integrated logis-tical planning for the simultaneous optimization ofvarious types of decisions, including facility locationand capacity, production, inventory, and distribution.

Çetinkaya et al.: An Integrated Outbound Logistics Model for Frito-Lay462 Interfaces 39(5), pp. 460–475, © 2009 INFORMS

However, despite the growing interest in this topic,our specific problem has not been previously ana-lyzed in the literature.Academic research has successfully documented

the cost savings associated with integrated decisionmaking for logistical planning. A detailed review ofthis general topic is beyond our scope. For relatedliterature, we refer the reader to Baita et al. (1998),Çetinkaya (2004), Keskin and Üster (2007), Sarmientoand Nagi (1999), and Thomas and Griffin (1996); werestrict our attention to previous research that specif-ically investigates the interaction between inventoryand transportation decisions. We note that the tradi-tional inventory models largely ignore this interac-tion because the early theory assumes that demandsshould be satisfied as soon as they arrive and thecustomer should cover delivery costs. Consequently,the outbound transportation costs are sunk costs(Hadley and Whitin 1963, pp. 17–18). However, thisis no longer true for private-fleet outbound distribu-tion in general and for VMI/D in particular, bothof which have received significant academic atten-tion during the past decade (Bertazzi et al. 2005,Çetinkaya and Lee 2000, Kleywegt et al. 2002, Toptaland Çetinkaya 2006).Specifically, two types of research emphasize the

interaction between inventory and transportationdecisions. The first type focuses on including trans-portation costs and (or) capacities in traditionalinventory problems without explicitly modelingtransportation-related (e.g., routing, consolidation,etc.) decision variables such as classical economicorder quantity, dynamic lot sizing, stochastic dynamicdemand, one-warehouse multiretailer, joint replenish-ment, and buyer-vendor coordination problems. It isworthwhile to note that the previous work on thistopic (Alp et al. 2003, Chan et al. 2002, Diaby andMartel 1993, Lee et al. 2003, Lipmann 1969, Toptaland Çetinkaya 2006, Üster et al. 2008, Van Eijs 1994)focuses on stylistic supply chain settings (e.g., singlevendor and single buyer, single vendor and multi-buyer, etc.) and neglects the vehicle-routing consider-ations that we address.The second set addresses the integration of inven-

tory and transportation decisions by explicitly con-sidering both inventory- and transportation-relateddecision variables. The literature on this latter type of

research is also abundant (Anily and Federgruen 1993,Baita et al. 1998, Çetinkaya 2004, Federgruen andSimchi-Levi 1995, Viswanathan and Mathur 1997).We note that our problem has several characteris-tics of the inventory-routing problem and its exten-sions (Archetti et al. 2007, Bertazzi 2008, Bertazzi et al.2005, Campbell and Savelsbergh 2004, Fumero andVercellis 1999, Kleywegt et al. 2002, Lei et al. 2006).The inventory-routing problem, studied by Campbelland Savelsbergh (2004), is a variation of the vehicle-routing problem in which the vendor can makedecisions about the timing, sizing, and routing ofdeliveries without allowing any shortages. However,most existing literature on this problem is method-oriented (e.g., large-scale mixed-integer programs); afew exceptions (e.g., Lei et al. 2006) focus on real-lifeapplications.Although the specific application setting in this

paper is similar to the inventory-routing problem, weconsider a more general supply chain setting than theprevious literature does. Our setting includes mul-tiple product groups and intermediate tiers; theserepresent the DCs and bins in which customers donot carry more than one period’s inventory and canreplenish from the DCs, bins, and directly from thefactory warehouse.

Problem Setting and FormulationWe consider a finite planning horizon in which eachperiod represents one week. Multiple DCs, bins,plants (that receive interplant shipments), and cus-tomers constitute the delivery locations on the out-bound supply chain. We model the tiers and flowsof the outbound supply chain (Figure 1). The flowsinclude (1) production output at the Irving plant;(2) finished-product inventories at the factory ware-house; (3) replenishment quantities from the factorywarehouse to the DCs and bins, plant-to-store ship-ment quantities to DD customers, and interplant ship-ment quantities to the other plants; (4) shipmentquantities from the DCs and bins to customers; and(5) customer demand that can be satisfied by ship-ments from DCs and bins and (or) direct supply fromthe factory warehouse. Our model’s objective is tominimize the handling, inventory, loading, routing,and delivery costs associated with the flows. Because


Staging/Staging/loadingloading

Bins (set B)Bins (set B)

Dijt

FactoryFactorywarehousewarehouse

For period t and product jFor period t and product j

Sdijt

Sbijt

Ibjt

DCs (set D)DCs (set D)

Idjt

Rbjt

Rdjt

Rojt

Iojt

CustomersCustomers(set I)(set I )

Pjt

TIER 4TIER 4 TIER 3TIER 3 TIER 2TIER 2 TIER 1TIER 1

Soijt

Rpjt

Irvingplant

Irvingplant

OtherplantsOtherplants

Figure 1: The figure illustrates our mathematical programming formulation, which represents the tiers and flowsof the underlying network and characterizes the outbound supply chain of the Irving plant.

our focus is outbound logistics, we do not model theproduction process in detail, and we omit produc-tion costs. However, we explicitly model the produc-tion capacities. One output of our model is a targetproduction output plan that is useful in developingdetailed production schedules.We introduce our notation and present the com-

plete mathematical formulation in the appendix. Toexplain our formulation, we next examine the tiers ofthe underlying supply chain from customers to theIrving plant and introduce the problem constraints.

Constraints Related to Inventory Decisionsfor Tiers 1–4Analyzing the chain in Figure 1 backwards, thefirst tier consists of the customers who generate thedemand flows that can be satisfied by shipping fromDCs and bins. For some customers, a DD can be sentfrom the factory warehouse; they can be served onthe same route with DCs and bins. To determine thesource(s) of supply for each customer, we use the cus-tomer’s DC or bin assignment and DD information.

Then, the demand constraints (1) and (2) dictate thatcustomer demand is satisfied by timely shipmentsto customers from DCs, bins, and the factory ware-house. These constraints also dictate that weekly totalshipments to a customer location do not exceed theweekly demand, because FLNA’s VMI/D programensures that customer locations never receive moreinventory than their weekly demand specifies.We note that we model the demand for each product

group at a customer location in a given week using anaggregate number of cartons. The actual cartons usedfor each product group are characterized by varyingsizes, i.e., storage volumes. To represent demand in anaggregate fashion, we define a cubic carton, i.e., a unitmeasure for our volume calculations, as the followingexample shows. Suppose that, for a product group, weuse two types of cartons with storage capacities of 0.5cubic feet and 1.5 cubic feet, respectively. Furthermore,let the usage proportions (historical data provided) forthe two carton types be 0.75 and 0.25, respectively. Wedefine a cubic carton for this product group with a


storage volume given by the weighted average of thevolumes of the carton types available for the productgroup: �0�75��0�5� + �0�25��1�5� = 0�75 cubic feet. Then,we compute the total demand for a product group ata customer location in a given week (in required cubicfeet) by multiplying the storage volume of the cubiccarton for the product group by the total number ofactual cartons (data provided).Next, considering the second tier of the chain, it

is straightforward to develop the inventory-balanceconstraints at the DCs (3) and bins (4). The vari-ables representing the replenishment quantities of theproducts sent from the factory warehouse to the DCsand bins, in successive periods, link the second (DCsand bins) and the third (staging and loading area ofthe factory warehouse) tiers of the chain in Figure 1.These variables also represent the truck loads that areen route from the factory warehouse to the DCs andbins in successive periods. Hence, we also use themto develop additional constraints that represent theloading and routing operations. We note that inven-tory levels at the DCs and bins should not violatethe capacity limits; hence, we have the correspondingstorage capacity constraints (5) and (6).Considering the flow conservation at the staging

and loading area of the factory warehouse, we havethe load balance constraints (7), which consider thetotal load from the factory warehouse to the DCs,bins, DD customers, and other plants. It is worth not-ing that, in these constraints, the interplant replenish-ment quantities are model parameters that representthe nonproduced item requirements at the otherplants. The variables representing the total loads ofthe products shipped from the factory warehouse,in successive periods, link the staging and loadingarea with the fourth tier, the factory warehouse. Con-sidering the replenishment (production) quantities ofthe products in successive periods, we also havethe corresponding inventory balance constraints atthe factory warehouse (8). The total inventory at thefactory warehouse should not violate the capacitytherein; therefore, we also have the storage capacityconstraints at the factory warehouse (9).Because we model the target production output lev-

els of different products in different periods as deci-sion variables, we say that our formulation translatescustomer demand into a target production output

plan. These target production output levels accountfor the loading and delivery schedules explicitly (wediscuss the loading and delivery considerations in thenext paragraph). When we have computed a numer-ical solution for the proposed model, we can usethe resulting target production output levels (of indi-vidual products in successive periods) as “translateddemand” values in the existing production planningand scheduling software. The target production out-put levels should not violate the aggregate capacityand product-based constraints (10) and (11).

Constraints Related to Truck Loadingand Routing at Tier 3Figure 2 illustrates the loading and routing operationsbetween the factory warehouse and the DCs, bins, DDcustomers, and other plants. We model these opera-tions explicitly using constraints (14)–(23).First, let us consider the loading and routing com-

ponent of the problem associated with the outboundshipments from the factory warehouse (Figure 2). Ourobjective is to minimize the outbound transportationcosts from the factory warehouse and to maximizetruck utilization. The OTR trucks based at the factorywarehouse are identical.The link between the routing operations and the

flow constraints is as follows. The load balance con-straints consider the total loads of the products to beshipped in successive periods from the factory ware-house to the delivery locations. The components of thetotal load shown on the left side of the load balanceconstraints represent the replenishment and shipmentquantities that should be loaded on trucks that areen route from the factory warehouse to the deliverylocations. Then, it is straightforward to verify the totalload and truck volume constraints (14) and drop-offconstraints (15)–(18). Note that the drop-off constraintsare associated with the set of delivery locations of therouting component of interest. The consequent truckarrival and departure constraints (constraints 19–22)assure that (1) the number of trucks arriving at a deliv-ery location is equal to the number of trucks leav-ing that location, and (2) all trucks leaving the factorywarehouse eventually return to this location. Any sub-set of the delivery locations can be on the route of anytruck. Hence, we must assure that the total load on atruck destined to visit any such subset does not exceed


1

z5ot = 1z45t = 1

z34t = 1

2

z23t = 1z12t = 1zo1t = 1

33

4

5

Staging and loadingStaging and loading Vehicle routingVehicle routing

12

4

5

zo1t = 1z12t = 1 z23t = 1

z34t = 1

z45t = 1z5ot = 1

Figure 2: The model’s loading and routing component deals with finding the subsets of delivery locations that willbe visited by the private fleet of trucks dispatched from the staging and loading area at the factory warehouse.

the truck volume; the ensuing route-based truck vol-ume constraints (23) follow.

Constraints Related to Market AreaDeliveries at Tier 2The final set of constraints is associated with smallertrucks (RD trucks) delivering indirect shipments tocustomers from the DCs and bins. At this level, wedifferentiate between the RD trucks based at the DCsand bins based on their distinct capacities. The setsof customers that each DC and bin serves are fixed.Each set represents a market area, which consists ofcustomers in close proximity and assigned to a spe-cific DC or bin. Under the VMI/D program, FLNAserves all customers in each market area using pre-set delivery routes. Hence, our main interest at thecustomer level is not load or route optimization; it isto assure that the truck volume constraints associatedwith deliveries from the DCs and bins are satisfied.

The two sets of dispatch capacity constraints (12)and (13) ensure that these requirements are satisfied.

Cost FunctionThe cost function has five specific components thatrepresent (1) touch costs (i.e., product-based per-unithandling costs at DCs and bins) associated with indi-rect customer deliveries via DCs and bins, (2) fixeddelivery costs to market areas from the DCs andbins, (3) holding costs at inventory-keeping locations,(4) fixed costs of loading and dispatching OTR trucksat the factory warehouse, and (5) distance-based rout-ing costs of OTR trucks leaving the factory ware-house. The first two terms of the objective functiontake into account the touch costs. A fixed deliverycost, which is associated with each dispatch from eachDC (bin) to its market area, is included in the thirdand fourth terms of the objective function. The fifth,sixth, and seventh terms correspond to the holding


costs at the inventory-keeping locations. Next, associ-ated with each OTR truck released from the factorywarehouse, we consider a fixed loading cost. Finally,we keep track of the number of drop-offs at eachdelivery location of the truck routes via the route-selection variables in successive periods; thus, we canincorporate the total mileage cost in the last term ofthe objective function.

Solution MethodologyAs we noted earlier, our solution methodologydecomposes the overall model into two subproblemsthat involve complementary inventory and routingcomponents. We illustrate the relationship betweenthe subproblems in the flowchart of our algorithm inthe appendix (Figure 8), and we also provide the spe-cific formulations of the subproblems. We solve thesubproblems iteratively until we cannot find a cost-based improvement for the overall solution or reachthe limit on maximum number of iterations.The inventory subproblem seeks to determine the

weekly replenishment and shipment quantities at theDCs, bins, and DD customers, and also considersthe requirements at the other plants, the touch costs,inventory holding costs, fixed delivery costs to mar-ket areas from the DCs and bins, and additional fixedcosts associated with loading and routing (i.e., theroute-based setup costs), as we describe in detailbelow. Given the weekly replenishment and shipmentquantities, the routing subproblem specifies the truckroutes and minimizes the actual loading and rout-ing costs. Our algorithm begins with an initializationmodule. Its main role is to set the replenishment andshipment quantities as input into the routing subprob-lem in the first iteration, as we describe next.

InitializationIn the initialization module, in which we addresseach product in each period, we set the shipmentquantities for each potential DD customer based onthat customer’s corresponding demand, to satisfy thedemand constraints (1). Hence, at this iteration, theDD customers do not receive any shipments fromtheir respective DCs or bins. We set the replenishmentquantities from the factory warehouse to the DCs,bins, and other plants based on the remaining over-all requirements of the corresponding locations and

using the inventory balance constraints (3) and (4).We specify the remaining overall requirements by theshipment quantities of the other customers, as dic-tated by the demand constraints (2) and the prod-uct requirements of the other plants. We also set thetotal load in a similar fashion using the load balanceconstraints (7).Utilizing the weekly DD, replenishment, and ship-

ment quantities—based either on the outcome of theinitialization module discussed above or the solutionof the inventory subproblem—in the routing subprob-lem, we determine the corresponding truck routeswhile minimizing the loading and routing costs ofthe OTR trucks based at the factory warehouse. Usingthe solution to the routing subproblem, we calcu-late the route-based setup costs associated with eachdelivery location of the routing subproblem in eachperiod. In the appendix, we describe how we calcu-late these costs and consider them as an input whilesolving the inventory subproblem. We now proceedwith a detailed discussion of the routing and inven-tory subproblems.

Routing SubproblemThe constraints and cost terms related to loading androuting operations in the overall model constitute therouting subproblem. Its main goal is to determine theOTR truck delivery routes from the factory warehouseto the delivery locations given the weekly replen-ishment and shipment quantities. A closer examina-tion of the formulation in the appendix reveals thatthe routing subproblem is separable for each period.Hence, it suffices to solve a routing problem succes-sively for each period (week) of the planning horizon.We use a savings algorithm, the Clarke and Wrightalgorithm (Clarke and Wright 1964, Chopra andMeindl 2001) with an additional step for improve-ment. When the savings algorithm terminates with aset of routes, we improve each generated route withmore than four delivery locations using a cheapestinsertion heuristic, a well-known travelling salesmanproblem heuristic.Before we solve a routing problem for each period,

we preprocess the given replenishment and shipmentquantities to see if full-truck load (FTL) shipments areneeded for a specific location; this might be the casefor major customers, such as Sam’s Club and Costco,and metropolitan area DCs. That is, we first process


FTL shipments so that the quantity leftover for eachlocation is less than the truck capacity. We note thateach FTL shipment constitutes a route with a sin-gle destination. We then specify the additional truckroutes and consider the leftover quantities, which arecalled less-than-truck load (LTL) shipments (each ona route with others).

Inventory SubproblemTraditional inventory models evaluate the trade-offsbetween fixed ordering and inventory holding costs todetermine replenishment quantities. In our inventorysubproblem, because of the influence of the routingsubproblem, the trade-off also includes touch costs atthe bins and DCs, fixed delivery costs to the mar-ket areas, and route-based setup costs dictated by theloading and routing costs of the OTR trucks based atthe factory warehouse. We discussed these cost com-ponents above, except for the route-based setup costs,which are an input from the routing subproblem.In each period, depending on its replenishment and

shipment quantity, a particular delivery location canbe visited by a truck(s) on a route(s). By sending atruck on a particular route, we incur a route-based,fixed-loading total-mileage cost. Each delivery loca-tion on the route is responsible for the correspondingcost. We allocate this cost to the delivery locations onthe route when calculating the corresponding route-based setup costs, as we describe in the appendix.Our proposed solution for the routing subproblem issuch that the set of routes of a given delivery loca-tion, l, includes at most one route in which l shares atruck with other delivery locations, and possibly otherroutes carrying FTL shipments to l (if the require-ments at l in a given period exceed the truck capac-ity). To ensure that the right delivery locations incurthe route-based setup costs in a given period, weuse binary variables. The impact of route-based setupcosts are reflected by the last four terms of the objec-tive function and the route-based setup constraintsof the inventory subproblem. These costs and con-straints establish the link between the inventory androuting subproblems.Except for the addition of the route-based setup

costs and constraints, the remaining formulationof the inventory problem is based on our overallmodel. That is, the inventory subproblem includesthe constraints associated with the demand, inventory

balance, storage capacity, load balance, productioncapacity, and dispatch capacity from the overallformulation, and the additional route-based setupconstraints. The objective function includes the corre-sponding route-based setup costs and all cost termsof the overall model, except the loading- and routing-related cost parameters considered in the routing sub-problem. We solve the inventory subproblem usingCPLEX 9.0 (CPLEX is a trademark of ILOG, Inc.).

Application ModulesOur application includes the database application andsolution algorithms, and provides broad functional-ity for optimization, strategy evaluation, and analy-sis of output files. The database application is usefulfor importing input data and processing the datato eliminate discrepancies in format and range. Theapplication also includes (1) a graphical user inter-face, (2) application programming interfaces (APIs)for constructing a network to facilitate the genera-tion of interdistances between the delivery locationsbased on the address information, and (3) modulesfor performing sensitivity analysis and report gener-ation (Figure 3). We designed the user interface andthe normalized database tables using Visual Basic forApplications and MS Access 2003 and implement oursolution using the C++ programming language, Con-cert Technology, and CPLEX 9.0. We perform the runson a computer with a 3 GHz Intel XEON processorand 6 GB RAM.The application’s information flow begins with

importing data using an interface that has a step-by-step procedure for performing the required data-import and data-integrity verification. The graphicaluser interface (Figure 4) allows the user to easilyimport, effectively manage, and efficiently query datapertaining to the logistics network, capacity, demand,and cost. The interface is also useful for selecting aset of potential DD customers based on a variety ofuser-specified criteria.The interdistance data are generated through the

API for input data, such as network-constructiondata, and are passed to the solution algorithms. TheAPI for solution algorithms provides formatted input,executes the algorithms, and reads the output solutioninto the database tables for later analysis and reportgeneration.


Figure 3: The graphic shows a snapshot of our application’s main menu.

Finally, the user can use the analysis module tochange the network, capacity, demand, and cost data,and to perform scenario analyses to evaluate newstrategies for incorporating DD customers for otherplants. The application requires little user training;in addition, it provides effective solutions in rea-sonable time and facilitates technology transfer foreasy implementation at FLNA plants. FLNA exec-utives have used it for distribution policy, strategyevaluation, and scenario analyses. Their main objec-tives have been to assess potential cost savings thatwould result from (1) direct deliveries for varyingdemand forecasts and (2) inventory optimization andreduction at the factory warehouse. A plant in Perry,Georgia has implemented the application for thesepurposes.

Benchmark Models andBenefits RealizedWe present our numerical results to demonstrateour approach’s effectiveness and benefits. First, wecompare our results to two benchmarks, the chase andcube-out policies. Neither considers DD customers.

The chase policy matches the demand by sendingthe exact weekly requirements (including the demandand safety stock) to the DCs and bins. Using thispolicy, we obtain a feasible solution to the inventorysubproblem by satisfying the replenishment and ship-ment requirements, and then solve the routing sub-problem. Hence, it provides a basis for measuring therelative gains associated with inventory optimizationand for considering DD customers. Using the cube-out policy, FTL shipments, which include the demandand some excess supply for the most popular prod-uct, potato chips, are sent to the DCs and bins toachieve FTL shipments. This benchmark, which aimsto increase truck utilization and routing efficiencywithout explicitly addressing the role of inventoryoptimization, represents the usual company practice.We consider 5 product groups that are produced

at Irving, 8 FLNA plants located elsewhere to whichinterplant shipments are sent, 8 DCs, 74 bins, 8,460customers with 85 potential DD customers, and12 weeks of demand data. We observe that our solu-tion for the optimization and benchmark models runsin a very reasonable time using real data. We obtain


Figure 4: The graphic shows a snapshot of the graphical user interface for network data of the factory warehouse.

the chase and cube-out policy solutions in a fewminutes and obtain a solution for our model within10 minutes. We also obtained similar run-time resultsfor a relatively larger problem at another FLNA plantin Perry, Georgia.In Table 1, we present the overall cost and its main

components for each model. In the last row of thistable, we report the percentage difference between thecost using our model and the benchmark models. Inthe following discussion, we elaborate on the detailedbenefits of our model relative to the two benchmarkmodels and the individual strengths of each model.

Detailed Analysis of Cost ComponentsIn Table 2, we show the cost-component results forour model and for each policy. Relative to the chasepolicy and cube-out policy, the inventory levels in ourmodel at all stock-keeping locations are significantlylower because of inventory optimization; the factorywarehouse achieves the greatest inventory reduction.

Compared to the two benchmark policies, the inven-tory holding costs at the factory warehouse in ourmodel are lower by as much as 76.92 percent (chasepolicy) and 73.62 percent (cube-out policy). The inven-tory holding costs at the DCs are lower by 24.52 per-cent and 25.59 percent compared to the chase policy

Our model Chase policy Cube-out($) ($) policy ($)

Touch cost 6�75 8�64 8�64Delivery cost (from DCs and bins) 33�77 45�17 45�17Inventory holding cost 2�83 4�08 4�34Loading cost (at the factory 28�63 28�30 25�72warehouse)

Mileage cost (from the factory 28�02 25�01 22�51warehouse)

Total cost 100 111�20 106�38

Our model is better by 11�20 6�38

Table 1: Examining the costs using our model and each policy indicatesthat our model outperforms both policies, and the cube-out policy outper-forms the chase policy.


Our model Chase policy Cube-out policy($) ($) ($)

Inventory holdingOverall 2�83 4�08 4�34Factory 0�24 1�04 0�91DCs 1�57 2�08 2�11Bins 1�02 0�97 1�31

TouchOverall 6�75 8�64 8�64DCs 4�50 6�19 6�19Bins 2�25 2�46 2�46

Deliveries from DCs and binsOverall 33�77 45�17 45�17DC 29�76 40�79 40�79Bin 4�01 4�37 4�37

Table 2: A detailed cost breakdown of the inventory subproblem for ourmodel and each policy is helpful in evaluating the impact of positioninginventory at the DCs, bins, or factory warehouse.

and cube-out policy, respectively. The inventory hold-ing costs at the bins decrease by 22.13 percent com-pared to the cube-out policy, whereas they increase byonly 4.90 percent compared to the chase policy.In terms of touch costs and delivery costs from

the DC and bins, the chase policy and cube-out pol-icy perform identically; our model outperforms bothbecause we explicitly consider DD customers. Ourmodel reduces the touch costs as much as 27.30 per-cent and 8.54 percent from the DCs and bins, respec-tively. Similarly, it reduces the total delivery costsfrom the DCs and bins by 27.04 percent and 8.54 per-cent, respectively. We note that higher cost reductionsin the DC-related touch and transportation costs arebecause of DD customer assignments. In general, DCsare responsible for serving large stores that are poten-tial DD customers. Using the chase and cube-out poli-cies, these large customers are usually served fromDCs, thus incurring higher touch and transportationcosts. Our model allows such customers to be serveddirectly from the factory warehouse. Next, we pro-ceed with a detailed discussion of additional benefitsassociated with explicitly considering DD customers.

Measuring the Value of Direct DeliveriesA closer examination of the results in Tables 1 and 2is useful for illustrating both the savings resultingfrom optimization and the explicit consideration ofDD customers.

One of the most striking results from the data inTable 1 is that, in each model, transportation-relatedexpenses (i.e., delivery costs associated with ship-ments from DCs and bins and loading and totalmileage costs associated with shipments from the fac-tory warehouse) represent approximately 90 percentof the cost. In our model, the loading costs at thefactory warehouse are higher by 1.16 percent and11.31 percent compared to the chase and cube-outpolicies, respectively. This is a consequence of theincreased number of delivery locations served fromthe factory warehouse (because of including DD cus-tomers). For the same reason, the total mileage costfrom the factory warehouse is higher by 12.03 percentand 24.48 percent compared to the chase and cube-out policies, respectively. However, in terms of overallcost, our model is better than the chase and cube-outpolicies by $11.20 and $6.38, respectively, because ourexplicit consideration of potential DD customers andinventory optimization helps to significantly reducethe touch costs, inventory holding costs, and deliverycosts from DCs and bins.As we note above, the potential DD customers

include 85 locations identified based on demand vol-ume. However, each potential DD customer does notreceive its supplies directly from the factory ware-house in each period. The selected number of DD cus-tomers varies between 66 and 74, as Figure 5 shows.For the specific planning horizon considered, in each

0

10

20

30

40

50

60

70

80

90

1 2 3 4 5 6 7 8 9 10 11 12Weeks

Num

ber

of c

usto

mer

s

Selected DD customersDD customers receiving FTL shipmentsPotential DD customers

Figure 5: When we consider a total of 85 potential DD customers, boththe number of selected DD customers and those receiving FTL shipmentsfrom the factory warehouse are significant.


70.0

40.0

50.0

60.0

20.0

30.0

0.0

10.0

1 2 3 4 5 6 7 8 9 10 11 12Weeks

%

LTL % FTL %

Figure 6: This illustration of the FTL and LTL shipments from the fac-tory warehouse to the selected DD customers, as a percentage of theirtotal demand, demonstrates that some customers still receive shipments,albeit in small amounts, via their respective DCs and bins.

week, most of the selected DD customers (varyingbetween 66.2 percent in week 2 and 98.6 percent inweek 7) receive FTL shipments from the factory ware-house. Shipping these large quantities directly, ratherthan via DCs and (or) bins, helps to save on touchcosts and inventory holding costs and also on deliverycosts from DCs and bins. For example, in week 7, 73of 74 selected DD customers receive FTL shipments.The touch, inventory holding, and delivery costs fromDCs and bins decrease by sending smaller quantitiesto the DCs and bins.We note that when a customer is identified as a DD

customer, that customer has the opportunity to receiveshipments directly from the factory warehouse; how-ever, this opportunity does not forbid shipments fromthe DCs and bins. In Figure 6, we illustrate the ship-ments to DD customers from the factory warehouse.Adding the FTL and LTL percentages sent from thefactory warehouse, we observe that 79.8 to 89.1 per-cent of the DD customer demand is shipped directlyfrom the factory warehouse. In Figure 6, the percent-age of FTL shipments varies between 48.4 and 61.3percent in different periods, and the percentage of LTLshipments varies between 27.8 and 35.5 percent.

Trade-Off Between Truck Utilizationand Inventory OptimizationWe next discuss the results of the trade-off betweentruck utilization, which FLNA considers a success cri-terion for efficient routing, and inventory optimiza-tion. As we mentioned above, the company uses thisprinciple for designing the routes by following thecube-out policy. One of its main concerns about our

Weeks1 2 3 4 5 6 7 98 10 11 12

90.0

80.0

70.0

60.0

50.0

40.0

30.0

20.0

10.0

%

BM2Our model

BM1

Figure 7: Using our model, average truck utilization is comparable to thatof the chase policy.

model was whether it could achieve high truck uti-lization. In Figure 7, we address this issue by showingthe truck utilization percentage for each model.As expected, the cube-out policy provides the high-

est truck utilization; it ranges between 81.8 and88.3 percent during the planning horizon; however,our model is comparable to the chase policy and per-forms well relative to company expectations. Basedon overall benefits, FLNA evaluates the model asproviding significant value to the company. In addi-tion to its cost advantage (Table 1), the model allowsFLNA to directly serve DD customers from the fac-tory warehouse with either FTL or LTL deliveries.Neither policy can provide this benefit because nei-ther can accommodate a route that includes DD cus-tomer locations.

ConclusionsOur research resulted in quantifiable benefits forFLNA. Both the proposed model and the solutionapproach are innovative because, despite the recentinterest of academia and industry in integrating logis-tical decisions, no published results exist that addressthe specific problem we discuss in this paper.A major goal of our research was inventory elimi-

nation at the Irving plant. To this end, FLNA had beeninvestigating the use of a new packing and shippingtechnology that would make direct supply from thefactory warehouse to DD customers a viable option.Our model provided input into FLNA’s decision mak-ing by evaluating the impact of DD customers.A careful investigation of the model’s results in

comparison to the chase and cube-out policies showedthat DD options, in addition to making efficient


routing and inventory decisions, eliminated inventoryat the factory warehouse. FLNA could also realize sig-nificant cost savings in comparison to the benchmarkmodels, which did not allow direct deliveries. Further-more, we were able to evaluate the criteria for specify-ing potential DD customers, the impact of positioninginventory at the DCs and bins versus the factory ware-house, and the trade-offs between truck utilizationand inventory optimization. An additional goal wasto evaluate the cube-out policy as implemented foroutbound shipments from plants to DCs. Our resultsrevealed that DDs from the Irving plant, when sup-ported by efficient routing and inventory replenish-ment decisions, presented a significant opportunity forcost savings compared to an estimate for the exist-ing system. When we implemented our model andthe two benchmark models for the FLNA plant inPerry, Georgia, we saw similar results. The studyand results received enthusiastic support from FLNA,and our collaboration continued with a closer exam-ination of the implications of operational (i.e., dailydelivery) requirements of the selected DD customers.Additional research, which we will document in afuture publication, is in progress.

Appendix

NotationWe use the following notation in our model formulation.

Sets and Indicest: time index, t = 1�2� � � � � T .J : set of product groups, j ∈ J .

D: set of DCs, d ∈ D.B: set of bins, b ∈ B.P : set of other plants receiving interplant ship-

ments, p ∈ P .I : set of customers, i ∈ I .

Di: set of DCs that serve customer i; i.e., Di �D.Bi: set of bins that serve customer i; i.e., Bi � B.o: factory warehouse.Io: set of potential DD customers; i.e., Io � I .

IDd : set of customers served by DC d; i.e., IDd � I .IBb : set of customers served by bin b; i.e., IBb � I .V : �o� ∪ D ∪ B ∪ P ∪ Io.E: links connecting the locations in V .

G�V �E�: road network that consists of vertices (locations)V and edges (links) E.

Demand ParametersDijt : demand (in cubic cartons) at customer i for product

j in period t.Rpjt : product requirement (shipment quantity) of plant p

for product j in period t sent from o.

Parameters Related to Production, Inventory, andTransportation Capacitiesdccapd : storage capacity at DC d.

bincapd : storage capacity at bin b.ocap: storage capacity at o.

pcapjt : production capacity (in cases) for product j inperiod t.

pj : production time coefficient of each case of prod-uct j .

apcapt : aggregate production capacity in t (in hours orminutes).

tcap: volume capacity of trucks based at o.tcapd : volume capacity of trucks based at DC d.tcapb : volume capacity of trucks based at bin b.

Cost Parameterssdj : per-unit touch cost of product j at DC d.sdj : per-unit touch cost of product j at bin b.ad : fixed delivery/dispatch cost of each truck leaving

DC d.ab : fixed delivery/dispatch cost of each truck leaving

bin b.hdj : per-unit-per-period inventory holding cost at DC d.hbj : per-unit-per-period inventory holding cost at bin b.hoj : per-unit-per-period inventory holding cost at o.A: fixed loading cost of each truck leaving o.

gmn: mileage cost between m and n for m�n ∈ V .

Decision VariablesSdijt : shipment quantity from DC d to customer i ∈ IDd of

product j in period t.Sbijt : shipment quantity from bin b to customer i ∈ IBb of

product j in period t.Soijt : DD quantity (from o) to customer i ∈ Io of product j

in period t.Idjt : beginning inventory of product j in period t at DC d.Ibjt : beginning inventory of product j in period t at bin b.

Rdjt : replenishment quantity of product j in period t sentto DC d from o.

Rbjt : replenishment quantity of product j in period t sentto bin b from o.

Rojt : total load of product j destined to the DCs and binsfrom o.

Iojt : beginning inventory of product j in period t at o.Pjt : target replenishment quantity of product j in period t

sent to o.Ct : number of dispatches in period t from o.ydt : number of drop-offs at DC d in period t.ybt : number of drop-offs at bin b in period t.ypt : number of drop-offs at plant p in period t.yit : number of drop-offs at customer i ∈ Io in period t by

trucks originating at o.zmnt : number of vehicles traveling from location m ∈ V to

location n ∈ V .xdt : number of trucks leaving DC d in period t.xbt : number of trucks leaving bin b in period t.


Additional Notation Related to SubproblemsKit : route-based setup cost at DD customer i ∈ Io in

period t.Kdt : route-based setup cost at DC d in period t.Kbt : route-based setup cost at bin b in period t.Kpt : route-based setup cost at plant p in period t.wit : binary indicator of route-based setup at DD customer

i ∈ Io in period t.wdt : binary indicator of route-based setup at DC d in

period t.wbt : binary indicator of route-based setup at bin b in

period t.wpt : binary indicator of route-based setup at plant p at

time t.M : a large number.

ModelWe formulate the overall problem as a mixed-integer pro-gram as follows.

minT∑

t=1

∑

j∈J

∑

d∈D

∑

i∈IDd

sjSdijt +T∑

t=1

∑

j∈J

∑

b∈B

∑

i∈IBb

sjSbijt

(Touch costs)

+T∑

t=1

∑

d∈D

adxdt +T∑

t=1

∑

b∈B

abxbt

(Delivery and dispatch costs)

+T∑

t=1

∑

j∈J

∑

d∈D

hdj Idjt +T∑

t=1

∑

j∈J

∑

b∈B

hbj Ibjt +T∑

t=1

∑

j∈J

hoj Iojt

(Inventory costs)

+ AT∑

t=1

Ct +T∑

t=1

∑

n∈V

∑

m∈V

gmnzmnt

(Loading and routing costs)

subject to Dijt =∑

d∈Di

Sdijt +∑

b∈Bi

Sbijt +Soijt ∀ i∈ Io� ∀ j ∈ J � ∀ t

(Demand), (1)

Dijt = ∑

d∈Di

Sdijt + ∑

b∈Bi

Sbijt ∀ i ∈ I\Io� ∀ j ∈ J � ∀ t

(Demand), (2)∑

i∈IDd

Sdijt = Idjt + Rdjt − Idj�t+1� ∀d ∈ D� ∀ j ∈ J � ∀ t

(Inv. bal. at DCs), (3)∑

i∈IBb

Sbijt = Ibjt + Rbjt − Ibj�t+1� ∀ b ∈ B� ∀ j ∈ J � ∀ t

(Inv. bal. at bins), (4)∑

j∈J

Idjt ≤ dccapd ∀d ∈ D� ∀ t

(Storage cap. at DCs), (5)

∑

j∈J

Ibjt ≤ bincapb ∀ b ∈ B� ∀ t

(Storage cap. at bins), (6)∑

i∈Io

Soijt + ∑

d∈D

Rdjt +∑

b∈B

Rbjt +∑

p∈P

Rpjt = Rojt

∀ j ∈ J � ∀ t (Load balance), (7)

Rojt = Iojt + Pjt − Ioj�t+1� ∀ j ∈ J � ∀ t

(Inv. bal. at o), (8)∑

j∈J

Iojt ≤ ocap ∀ t (Storage cap. at o), (9)

∑

j∈J

pjPjt ≤ apcapt ∀ t

(Aggregate prodn. cap.), (10)

Pjt ≤ pcapjt ∀ j ∈ J � ∀ t

(Product-based prodn. cap.), (11)

xdt ≥∑

i∈IDd

∑j∈J Sdijt

tcapd

∀d ∈ D� ∀ t

(Dispatch cap. for DCs), (12)

xbt ≥∑

i∈IBb

∑j∈J Sbijt

tcapb

∀ b ∈ B� ∀ t

(Dispatch cap. for bins), (13)

Ct ≥∑

j∈J Rojt

tcap∀ t

(Total load and truck volume), (14)

ydt ≥∑

j∈J Rdjt

tcap∀d ∈ D� ∀ t

(Drop-offs at DCs), (15)

ybt ≥∑

j∈J Rbjt

tcap∀ b ∈ B� ∀ t

(Drop-offs at bins), (16)

ypt ≥∑

j∈J Rpjt

tcap∀p ∈ P� ∀ t

(Drop-offs at other plants), (17)

yit ≥∑

j∈J Soijt

tcap∀ i ∈ Io� ∀ t

(Drop-offs at DD customers), (18)∑

m∈V

zmnt = ynt ∀n ∈ V \�o�� ∀ t

(Arrivals to Tier 2), (19)∑

n∈V

zmnt = ymt ∀m ∈ V \�o�� ∀ t

(Departures from Tier 2), (20)


∑

m∈V

zmot = Ct ∀ t (Arrivals to o), (21)

∑

n∈V

zont = Ct ∀ t (Departures from o), (22)

∑

m�S

∑

n∈S

zmnt ≥∑

m∈S

∑j∈J Rmjt

tcap∀S ⊆ V \�o�� ∀ t

(Route-based truck volume). (23)

Nonnegative Integer Variables

Sdijt ∀d ∈ D� i ∈ I� j ∈ J � ∀ t�

Sbijt ∀ b ∈ B� i ∈ I� j ∈ J � ∀ t�

Soijt ∀ i ∈ Io� j ∈ J � ∀ t�

Rdjt� Idjt� Rbjt� Ibjt ∀d ∈ D� b ∈ B� j ∈ J � ∀ t�

Rojt� Iojt� Pjt ∀ j ∈ J � ∀ t�

xdt ∀d ∈ D� ∀ t�

xbt ∀ b ∈ B� ∀ t�

Ct ∀ t�

ydt ∀d ∈ D� ∀ t�

ybt ∀ b ∈ B� ∀ t�

yit ∀ i ∈ Io� ∀ t�

zmnt ∀m ∈ V � n ∈ V � ∀ t�

Obtaining an exact optimal solution is impractical for theoverall model because of its size and complexity. We over-come this computational difficulty by developing a heuristicthat builds on the decomposition of the above model intotwo subproblems involving inventory and routing compo-nents. We first describe these subproblems and then presenta flowchart of our iterative heuristic.

Routing SubproblemInput parameters are the inventory-related decision vari-ables Soijt , Rdjt , Rbjt , and Rojt and input parameters A, gmn,Rpjt , and tcap. The output includes (1) routing-related deci-sion variables Ct , ydt , ybt , ypt , yit , and zmnt , and (2) setupcosts Kdt , Kbt , Kpt , and Kit .

min AT∑

t=1

Ct +T∑

t=1

∑

n∈V

∑

m∈V

gmnzmnt

subject to constraints (14)–(23) with variables Ct , ydt , ybt ,yit , and zmnt as defined above.

When we obtain a solution, we calculate the route-basedsetup costs of the delivery locations on the resulting routes.For a given delivery location l ∈ V \�o�, let RouteSetlt denotethe set of routes to which delivery location l belongs in

period t. Also, let rs� denote the number of delivery loca-tions on route � ∈ RouteSetlt . Then, the route-based setupcost of delivery location l in period t is

Klt = ∑

�∈RouteSetlt

A +∑n∈�

∑m∈� gmnzmnt

rs�

�

We explicitly consider it in the subsequent inventorysubproblem.

Inventory SubproblemInput parameters include the route-based setup costsdenoted by Kit , Kdt , Kbt , and Kpt for i ∈ Io, d ∈ D, b ∈ B, p ∈ P ,and t = 1� � � � � T . We consider these costs and inventory-related cost parameters sj , hdj , hbj , and hoj , loading-relatedcost and capacity parameters ad , ab , tcapd , and tcapb , cus-tomer and interplant demand parameters Dijt and Rpjt , andinitial and end-of-horizon inventory levels at the inventory-keeping locations Idj0, Ibj0, Ioj0, Idj�T +1�, Ibj�T +1�, and Ioj�T +1�. Theoutput includes (1) all inventory-related decision variablesSdijt , Sbijt , Soijt , Idjt , Ibjt , Iojt , Rdjt , Rbjt , and Rojt ; (2) the numberof trucks required at DCs, xdt , and bins, xbt , for customerdeliveries; and (3) binary route-based setup indicators wit ,wdt , wbt , and wpt .

minT∑

t=1

∑

j∈J

∑

d∈D

∑

i∈IDd

sjSdijt +T∑

t=1

∑

j∈J

∑

b∈B

∑

i∈IBb

sjSbijt

+T∑

t=1

∑

d∈D

adxdt +T∑

t=1

∑

b∈B

abxbt +T∑

t=1

∑

j∈J

∑

d∈D

hdj Idjt

+T∑

t=1

∑

j∈J

∑

b∈B

hbj Ibjt +T∑

t=1

∑

j∈J

hoj Iojt +T∑

t=1

∑

i∈Io

Kitwit

+T∑

t=1

∑

d∈D

Kdtwdt +T∑

t=1

∑

b∈B

Kbtwbt +T∑

t=1

∑

p∈P

Kptwpt

subject to constraints (1)–(13)� and∑

j∈J Soijt

M≤ wit ∀ i ∈ Io� ∀ t

(Route-based setup at DD customers), (24)∑

j∈J Rdjt

M≤ wdt ∀d ∈ D� ∀ t

(Route-based setup at DCs), (25)∑

j∈J Rbjt

M≤ wbt ∀ b ∈ B� ∀ t

(Route-based setup at bins), (26)∑

j∈J Rpjt

M≤ wpt ∀p ∈ P� ∀ t

(Route-based setup at plants), (27)

with variables Sdijt , Sbijt , Soijt , Rdjt , Rojt , xdt , and xbt as definedabove and binary integer variables wnt ∀n ∈ V \�o�� ∀ t.


Initialization

Routing subproblem

costNew = cost(INV) + cost(VRP)

No

Sojt, Rdjt, Rbjt, Rpjt, Rojt

Yes

cost = costNew

Route-based setup costs: Kit, Kdt, Kbt, Kpt

iterNo = iterNo + 1

Inventory subproblem

cost – costNewcostNew

> 0.01

Sojt, Rdjt, Rbjt, Rpjt, Rojt

StopiterNo > MaxIterNo

Figure 8: The flowchart shows the solution algorithm in which cost(VRP)and cost(INV), respectively, represent the corresponding objective func-tion values of these two subproblems.

Solution AlgorithmThe algorithm involves iteratively solving the routing andinventory subproblems described above, which we outlinein Figure 8.

AcknowledgmentsThis research was supported by Frito-Lay and PepsiCounder contract C05-00582 and Texas Engineering Exper-iment Station Proposal 05-1073, during August 2005 toFebruary 2006. We particularly thank Greg Bellon, RobertW. Mackley, and Larry Sweet of Frito-Lay for their supportand input throughout the project.

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